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How to Extend Your Playing Time

30 December 2003

Besides money and the all-you-can-eat buffet, folks flock to casinos with their sights set on staying in action so they don't have to pretend there's something to do other than gamble and dine. Too, endurance offers a chance to win more when ahead, come back if behind, and rack up credits toward those coveted comps.

Nobody can predict whether any particular session will end in a rapid rout or run on indefinitely. But it's possible to find the precise chance to survive any stated number of rounds without depleting a specified stake, knowing edge, average bet size, and characteristic round-to-round bankroll fluctuations. Of course, it takes more than a moderate mastery of some rather messy math.

Even then, the precision isn't all that meaningful because solid citizens typically tune their play to prevailing conditions and can only guess what the edge, bet size, or volatility will be during a session. Fortunately, there are simplifications that can be used to approximate the extent to which changes in the various factors influence the likelihood of survival. (For purists, what follows was derived by solving the rigorous equation under various conditions, then tabulating results and using multiple regression to get linear coefficients for the variables.)

Edge: For every 1 percent increase or decrease in house edge, the chance of surviving with all else held constant moves in the opposite direction by 2.5 percent. That is, if you're playing craps and switch money from flat line bets with 1.4 percent edge to high odds at 0.4 percent overall, the 1 percent drop in house advantage improves your probability of being in action for a desired period by roughly 2.5 percent.

Bet and bankroll: Divide average wager by bankroll size. For instance, betting \$10 per round on a \$1,000 poke gives 10/1000 or 0.01; with \$25 bets and a \$1,500 stake, the quotient is 0.0167. Subtract the new value from the old if you change your bet, bankroll, or both, and multiply the result by -700 to get the effect on prospects for survival expressed directly as a percentage. With the figures cited, the bet/bankroll ratio rises by 0.0067. Multiplying by -700 and rounding to one decimal place, your chance of remaining in the game is estimated to fall by 4.7 percent.

Volatility: This critical but obscure quantity depends on chances of winning and associated payoffs. Analysts measure it using "standard deviation," which you can interpret as the nominal bankroll change during a round, for every dollar bet. A few touchstones will let you make educated guesses about its value for most table games. Even-money bets: 1.0; blackjack: 1.1; placing the four at craps: 1.3; placing the five at craps: 1.2; single spots at roulette: 5.8. If you switch bets, subtract the old standard deviation from the new and multiply by -10 to get the impact stated as a percentage. Going from the four to the five at craps raises your chance of surviving by -10 x (1.2 - 1.3) = 1 percent. Changing from a number such as 31 to an outside bet like red at roulette, same amount, improves your shot by -10 x (1.0 - 5.8) = 48 percent. Varying bets during play also raises volatility. For instance, a shift from \$10 flat to \$5 half the time, and \$10 and \$25 a quarter of the time each, raises volatility in an even-money game from 1.0 to 1.2. This cuts the probability of survival by -10 x (1.2 - 1.0) or 2 percent.

Rounds: For every additional hundred rounds you set as your target session duration, your chances of fulfillment drop by 4 percent, and conversely. Say you normally play roulette for two hours, about 100 spins. But you think you can get a better comp if you put in another hour. You're looking at 150 spins. The extra 50 rounds cut 2 percent from the chance that your bankroll will carry you through.

So, inquiring minds might want to know how best to extend playing time? Shop for or play to get lower edge, bet closer to flat and avoid longshots, and either lower your wagers or raise your bankroll. And, with the multipliers presented, you can easily estimate how much any or all of these tactics should help. Useful, because as the poet, Sumner A Ingmark, memorably mused:

'Tween commonplace and esoteric,
Lies cognizance of facts numeric.